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Tempered Distributions and the Fourier Transform

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Tempered Distributions and the Fourier Transform CHAPTER 1 Tempered distributions and the Fourier transform Microlocal analysis is a geometric theory of distributions, or a theory of geomet- ric distributions. Rather than study general distributions { which are like general continuous functions but worse { we consider more specific types of distributions which actually arise in the study of differential and integral equations. Distri- butions are usually defined by duality, starting from very \good" test functions; correspondingly a general distribution is everywhere \bad". The conormal dis- tributions we shall study implicitly for a long time, and eventually explicitly, are usually good, but like (other) people have a few interesting faults, i.e. singulari- ties. These singularities are our principal target of study. Nevertheless we need the general framework of distribution theory to work in, so I will start with a brief in- troduction. This is designed either to remind you of what you already know or else to send you off to work it out. (As noted above, I suggest Friedlander's little book [4] - there is also a newer edition with Joshi as coauthor) as a good introduction to distributions.) Volume 1 of H¨ormander'streatise [8] has all that you would need; it is a good general reference. Proofs of some of the main theorems are outlined in the problems at the end of the chapter. 1.1. Schwartz test functions To fix matters at the beginning we shall work in the space of tempered distribu- tions. These are defined by duality from the space of Schwartz functions, also called the space of test functions of rapid decrease. We can think of analysis as starting off from algebra, which gives us the polynomials. Thus in Rn we have the coordi- nate functions, x1; : : : ; xn and the constant functions and then the polynomials are obtained by taking (finite) sums and products: X α n (1.1) φ(x) = pαx ; pα 2 C; α 2 N0 ; α = (α1; : : : ; αn); jα|≤k n α α1 αn Y αj where x = x1 : : : xn = xj and N0 = f0; 1; 2;::: g: j=1 A general function φ : Rn −! C is differentiable atx ¯ if there is a linear function n P `x¯(x) = c + (xj − x¯j)dj such that for every > 0 there exists δ > 0 such that j=1 ¯ (1.2) jφ(x) − `x¯(x)j ≤ jx − x¯j 8 jx − x¯j < δ: The coefficients dj are the partial derivative of φ at the pointx: ¯ Then, φ is said to be differentiable on Rn if it is differentiable at each pointx ¯ 2 Rn; the partial derivatives are then also functions on Rn and φ is twice differentiable if the partial 13 14 1. TEMPERED DISTRIBUTIONS AND THE FOURIER TRANSFORM derivatives are differentiable. In general it is k times differentiable if its partial derivatives are k − 1 times differentiable. If φ is k times differentiable then, for eachx ¯ 2 Rn; there is a polynomial of degree k; X jαj α pk(x;x ¯) = aαi (x − x¯) /α!; jαj = α1 + ··· + αn; jα|≤k such that for each > 0 there exists δ > 0 such that k (1.3) jφ(x) − pk(x; x¯)j ≤ jx − x¯j if jx − x¯j < δ: Then we set α (1.4) D φ(¯x) = aα: If φ is infinitely differentiable all the Dαφ are infinitely differentiable (hence con- tinuous!) functions. Definition 1.1. The space of Schwartz test functions of rapid decrease consists n n of those φ : R −! C such that for every α; β 2 N0 (1.5) sup jxβDαφ(x)j < 1; n x2R it is denoted S(Rn): From (1.5) we construct norms on S(Rn): α β (1.6) kφkk = max sup jx D φ(x)j: jαj+jβ|≤k n x2R It is straightforward to check the conditions for a norm: (1) kφkk ≥ 0; kφkk = 0 () φ ≡ 0 (2) ktφkk = jtjkφkk; t 2 C n (3) kφ + kk ≤ kφkk + k kk 8 φ, 2 S(R ): The topology on S(Rn) is given by the metric X kφ − kk (1.7) d(φ, ) = 2−k : 1 + kφ − kk k See Problem 1.4. Proposition 1.1. With the distance function (1.7), S(Rn) becomes a complete metric space (in fact it is a Fr´echetspace). Of course one needs to check that S(Rn) is non-trivial; however one can easily see that 2 n (1.8) exp(−|xj ) 2 S(R ): In fact there are lots of smooth functions of compact support and 1 n n n (1.9) Cc (R ) = fu 2 S(R ); u = 0 in jxj > R = R(u)g ⊂ S(R ) is dense. The two elementary operations of differentiation and coordinate multiplication give continuous linear operators: n n xj : S(R ) −! S(R ) (1.10) n n Dj : S(R ) −! S(R ): 1.2. LINEAR TRANSFORMATIONS 15 Other important operations we shall encounter include the exterior product, n m n+m S(R ) × S(R ) 3 (φ, ) 7! φ 2 S(R ) (1.11) φ (x; y) = φ(x) (y): k n and pull-back or restriction. If R ⊂ R is identified as the subspace xj = 0; j > k; then the restriction map ∗ n k ∗ (1.12) πk : S(R ) −! S(R ); πkf(y) = f(y1; : : : ; yk; 0;:::; 0) is continuous (and surjective). 1.2. Linear transformations A linear transformation acts on Rn as a matrix (This is the standard action, n but it is potentially confusing since it means that for the basis elements ej 2 R ; n P Lej = Lkjek:) k=1 n n n X (1.13) L : R −! R ; (Lx)j = Ljkxk: k=1 The Lie group of invertible linear transformations, GL(n; R) is fixed by several equivalent conditions L 2 GL(n; R) () det(L) 6= 0 −1 −1 n (1.14) () 9 L s.t. (L )Lx = x 8 x 2 R −1 n () 9 c > 0 s.t. cjxj ≤ jLxj ≤ c jxj 8 x 2 R : Pull-back of functions is defined by L∗φ(x) = φ(Lx) = (φ ◦ L)(x): The chain rule for differentiation shows that if φ is diffferentiable then (So Dj transforms as a basis of Rn as it should, despite the factors of i:) n ∗ X ∗ (1.15) DjL φ(x) = Djφ(Lx) = Lkj(Dkφ)(Lx) = L ((L∗Dj)φ)(x); k=1 n X L∗Dj = LkjDk: k=1 From this it follows that ∗ n n (1.16) L : S(R ) −! S(R ) is an isomorphism for L 2 GL(n; R): To characterize the action of L 2 GL(n; R) on S0(Rn) consider, as usual, the distribution associated to L∗φ : Z (1.17) TL∗φ( ) = φ(Lx) (x)dx n R Z −1 −1 −1 −1 ∗ = φ(y) (L y)j det Lj dy = Tφ(j det Lj (L ) ): n R 16 1. TEMPERED DISTRIBUTIONS AND THE FOURIER TRANSFORM Since the operator j det Lj−1(L−1)∗ is an ismorphism of S(Rn) it follows that if we take the definition by duality ∗ −1 −1 ∗ 0 n n (1.18) L u( ) = u(j det Lj (L ) ); u 2 S (R ); 2 S(R );L 2 GL(n; R) ∗ 0 n 0 n =) L : S (R ) −! S (R ) is an isomorphism which extends (1.16) and satisfies (1.19) ∗ ∗ ∗ ∗ ∗ 0 n DjL u = L ((L∗Dj)u);L (xju) = (L xj)(L u); u 2 S (R );L 2 GL(n; R); as in (1.15). 1.3. Tempered distributions As well as exterior multiplication (1.11) there is the even more obvious multi- plication operation n n n S(R ) × S(R ) −! S(R ) (1.20) (φ, ) 7! φ(x) (x) which turns S(Rn) into a commutative algebra without identity. There is also integration Z n (1.21) : S(R ) −! C: Combining these gives a pairing, a bilinear map Z n n (1.22) S(R ) × S(R ) 3 (φ, ) 7−! φ(x) (x)dx: n R If we fix φ 2 S(Rn) this defines a continuous linear map: Z n (1.23) Tφ : S(R ) 3 7−! φ(x) (x)dx: Continuity becomes the condition: n (1.24) 9 k; Ck s.t. jTφ( )j ≤ Ckk kk 8 2 S(R ): We generalize this by denoting by S0(Rn) the dual space, i.e. the space of all con- tinuous linear functionals 0 n n u 2 S (R ) () u : S(R ) −! C n 9 k; Ck such that ju( )j ≤ Ckk kk 8 2 S(R ): Lemma 1.1. The map n 0 n (1.25) S(R ) 3 φ 7−! Tφ 2 S (R ) is an injection. n R 2 Proof. For any φ 2 S(R );Tφ(φ) = jφ(x)j dx; so Tφ = 0 implies φ ≡ 0: If we wish to consider a topology on S0(Rn) it will normally be the weak topol- ogy, that is the weakest topology with respect to which all the linear maps 0 n n (1.26) S (R ) 3 u 7−! u(φ) 2 C; φ 2 S(R ) are continuous. This just means that it is given by the seminorms n (1.27) S(R ) 3 u 7−! ju(φ)j 2 R 1.4. TWO BIG THEOREMS 17 where φ 2 S(Rn) is fixed but arbitrary. The sets 0 n (1.28) fu 2 S (R ); ju(φj)j < j; φj 2 Φg form a basis of the neighbourhoods of 0 as Φ ⊂ S(Rn) runs over finite sets and the j are positive numbers. Proposition 1.2. The continuous injection S(Rn) ,!S0(Rn); given by (1.25), has dense range in the weak topology.
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